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arXiv:0705.0371v1 [cond-mat.stat-mech] 2 May 2007

Magneto-elastic waves in crystals of magnetic molecules

C. Calero, E. M. Chudnovsky, and D. A. Garanin

Department of Physics and Astronomy,

Lehman College, City University of New York,

250 Bedford Park Boulevard West, Bronx, New York 10468-1589, U.S.A.

February 1, 2008

We study magneto-elastic effects in crystals of magnetic molecules. Coupled equations of motion

for spins and sound are derived and the possibility of strong resonant magneto-acoustic coupling

is demonstrated. Dispersion laws for interacting linear sound and spin excitations are obtained

for bulk and surface acoustic waves. We show that ultrasound can generate inverse population of

spin levels. Alternatively, the decay of the inverse population of spin levels can generate ultrasound.

Possibility of solitary waves of the magnetization accompanied by the elastic twists is demonstrated.

PACS numbers: 75.50.Xx, 73.50.Rb, 75.45.+j

I. INTRODUCTION

Crystals of molecular magnets are paramagnets1,2that

have the ability to maintain macroscopic magnetization

for a long time in the absence of the external magnetic

field. This is a consequence of the magnetic bi-stability

of individual molecules3that in many respects behave as

superparamagnetic particles. The latter is due to a large

spin (e.g., S = 10 for Mn-12 and Fe-8) and high magnetic

anisotropy of the molecules. Together with quantization

of spin energy levels this leads to a distinctive feature

of molecular magnets: A staircase hysteresis curve in a

macroscopic measurement of the magnetization.4

Hybridization of electron paramagnetic resonance

(EPR) with longitudinal ultrasonic waves has been stud-

ied by Jacobsen and Stevens5within a phenomenologi-

cal model of magneto-elastic interaction proportional to

the magnetic field.General theory of magneto-elastic

effects on the phonon dispersion and the sound veloc-

ity in conventional paramagnets has been developed by

Dohm and Fulde.6The advantage of molecular magnets

is that they, unlike conventional paramagnets, can be

prepared in a variety of magnetic states even in the ab-

sence of the magnetic field. Spontaneous transitions be-

tween spin levels in molecular magnets are normally due

to the emission and absorption of phonons. Interactions

of molecular spins with phonons have been studied in the

context of magnetic relaxation,7,8,9,10conservation of an-

gular momentum,11,12,13phonon Raman processes,14and

phonon superradiance.15Parametric excitation of acous-

tic modes in molecular magnets has been studied.16,17

It has been suggested that surface acoustic waves can

produce Rabi oscillations of magnetization in crystals of

molecular magnets.18In this paper we study coupled dy-

namics of paramagnetic spins and elastic deformations at

a macroscopic level.

When considering magneto-elastic waves in paramag-

nets the natural question is why the adjacent spins should

rotate in unison rather than behave independently. In

ferromagnets the local alignment of spins is due to the

strong exchange interaction. Due to this interaction the

length of the local magnetization is a constant through-

out the ferromagnet. We shall argue now that a some-

what similar quantum effect exists in a system of weakly

interacting two-level entities described by a fictitious spin

1/2. Indeed, since any product of Pauli matrices reduces

to a single Pauli matrix σα, interaction of N independent

two-state systems with an arbitrary field A(r) should be

linear on σα,

H =

N

?

n=1

gαβσ(n)

αAβ(rn) ,(1)

where σ(n)describes a two-state system located at a

point r = rn. If A was independent of coordinates, then

the Hamiltonian (1) would reduce to

H = gαβΣαAβ, (2)

where

Σ =

N

?

n=1

σ(n)

(3)

is the total fictitious spin of the system. In this case

the interaction Hamiltonian would commute with Σ2,

thus preserving the length of the total fictitious “mag-

netization”. This observation is crucial for understand-

ing Dicke superradiance:19A system of independent two-

state entities behaves collectively in a field whose wave-

length significantly exceeds the size of the system. When

the wavelength of the field is small compared to the size

of the system but large compared to the distance be-

tween the two-state entities, the same argument can be

made about the rigidity of Σ =?σ(n)summed up over

Consequently, the system that has been initially prepared

in a state with all spins up, and then is allowed to evolve

through interaction with a long-wave Bose field, should

conserve the length of the local “magnetization” in the

same way as ferromagnets do.

The relevance of the above argument to the dynam-

ics of magnetic molecules interacting with elastic defor-

mations becomes obvious when only two spin levels are

the distances that are small compared to the wavelength.

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important. This is the case when the low-energy dynam-

ics of the molecular magnet is dominated by, e.g., tunnel

split spin-levels or when the magneto-acoustic wave is

generated by a pulse of sound of resonant frequency. Re-

cently, experiments with surface acoustic waves in the

GHz range have been performed in crystals of molecu-

lar magnets.20The existing techniques, in principle, al-

low generation of acoustic frequencies up to 100 GHz.21

This opens the possibility of resonant interaction of gen-

erated ultrasound with spin excitations. In this paper we

study coupled magneto-elastic waves in the ground state

of a crystal of molecular magnets. We derive equations

describing macroscopic dynamics of sound and magne-

tization and show that high-frequency ultrasound inter-

acts strongly with molecular spins when the frequency of

the sound equals the distance between spin levels. We

obtain the dispersion relation for magneto-elastic waves

and show that non-linear equations of motion also pos-

sess solutions describing solitary waves of magnetization

coupled to the elastic twists.

The paper is organized as follows. The model of spin-

phonon coupling is discussed in Section II where coupled

magneto-elastic equation are derived. Linear magneto-

elastic waves are studied in Section III where we obtain

dispersion laws for bulk and surface acoustic waves. Non-

linear solitary waves are studied in Section IV. Sugges-

tions for experiments are made in Section V.

II.MODEL OF MAGNETO-ELASTIC

COUPLING

We consider a molecular magnet interacting with

a local crystal field described by a phenomenological

anisotropy Hamiltonianˆ HA. The spin cluster is assumed

to be more rigid than its elastic environment, so that the

long-wave crystal deformations can only rotate it as a

whole but cannot change its inner structure responsible

for the parameters of the Hamiltonianˆ HA. This approx-

imation should apply to many molecular magnets as they

typically have a compact magnetic core inside a large unit

cell of the crystal. In the presence of deformations of the

crystal lattice, given by the displacement field u(r), local

anisotropy axes defined by the crystal field are rotated

by the angle

δφ(r,t) =1

2∇ × u(r,t).

As a consequence of the full rotational invariance of the

system (spins + crystal lattice), the rotation of the lattice

is equivalent to the rotation of the operatorˆS in the

opposite direction, which can be performed by the (2S +

1) × (2S + 1) matrix in the spin space,13

ˆS →ˆR−1ˆSˆR,

Therefore, the total Hamiltonian of a molecular magnet

in the magnetic field B must be written as

(4)

ˆR = eiˆS·δφ.(5)

ˆ H = e−iˆS·δφ ˆ HAeiˆS·δφ+ˆ HZ+ˆ Hph,(6)

whereˆ HAis the anisotropy Hamiltonian in the absence

of phonons,ˆ HZ= −gµBB·ˆS is the Zeeman Hamiltonian

andˆ Hphis the Hamiltonian of harmonic phonons. The

angle of rotation produced by the deformation of the lat-

tice is small, so one can expand Hamiltonian (6) to first

order in the angle δφ and obtain

ˆ H ≃ˆ H0+ˆ Hs−ph,(7)

whereˆ H0is the Hamiltonian of non-interacting spins and

phonons

ˆ H0=ˆ HS+ˆ Hph,

andˆ Hs−phis the spin-phonon interaction term, given by

?

ˆ HS=ˆ HA+ˆ HZ,(8)

ˆ Hs−ph= i

ˆ HA,ˆS

?

· δφ.(9)

A. Coupling of spins to the elastic twists

For certainty, we consider a crystal of molecular mag-

nets with the anisotropy Hamiltonian

ˆ HA= −DˆS2

z+ˆV ,(10)

whereˆV is a small term that does not commute with

theˆSzoperator. This term is responsible for the tunnel

splitting, ∆, of the levels on resonance.

Atlow temperature and

kBT,gµBB ? ∆, when the frequency of the displace-

ment field u(r) satisfies ω ≪ 2DS/?, only the two lowest

states ofˆ HA are involved in the evolution of the sys-

tem. Thus, one can reduce the spin-Hamiltonian of the

molecular magnet to an effective two-state Hamiltonian

in terms of pseudospin-1/2 operators ˆ σi,

smallmagneticfield,

ˆ H(eff)

S

= −1

2(Wez+ ∆ex) · ˆ σ ,(11)

where ˆ σi are the Pauli matrices in the basis of theˆSz-

states close to the resonance between |S? and |−S?, and

W = ES− E−Sis the energy difference for the resonant

states at ∆ = 0. The non-degenerate eigenfunctions of

ˆ H(eff)

1

√2(C±|S? ∓ C∓| − S?)

S

are

|ψ∓? =

(12)

with

C±=

?

1 ±

W

√∆2+ W2. (13)

In terms of |ψ∓? the Hamiltonian (11) can be written as

= −1

2

ˆ H(eff)

S

?

W2+ ∆2 ˆ˜ σz,(14)

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whereˆ˜ σi are now the Pauli matrices in the new basis

|ψ±?, i.e.,ˆ˜ σz = |ψ+??ψ+| − |ψ−??ψ−|. The projection

of the spin-phonon interaction Hamiltonian (9) onto this

new two-state basis results in

ˆ H(eff)

s−ph=

?

i,j=±

?ψi|ˆ Hs−ph|ψj?|ψi??ψj| = δφzS∆ˆ˜ σy,

(15)

withˆ˜ σy= −i|ψ+??ψ−| + i|ψ−??ψ+|. The total Hamilto-

nian (6) of a single molecular magnet becomes

ˆ H(eff)= −1

b(eff)=

2b(eff)·ˆ˜ σ +ˆ Hph,

?

W2+ ∆2ez− 2δφzS∆ey.(16)

Here we have assumed that the perturbation introduced

by the spin-phonon interaction is much smaller than the

perturbationˆV producing the splitting ∆, which will usu-

ally be the case. Note also that ∆ and W can in gen-

eral be made r-dependent to account for possible inho-

mogeneities of the crystal.

When considering magneto-elastic excitations we will

need to know whether they are accompanied by a non-

zero local magnetization of the crystal. For that rea-

son it is important to have the magnetic moment of the

molecule,

mz= gµB?Sz?,(17)

(with g being the gyromagnetic ratio and µB being the

Bohr magneton), in terms of its wave function

|Ψ? = K+|ψ+? + K−|ψ−?,(18)

where K± are arbitrary complex numbers satisfying

|K−|2+|K+|2= 1. With the help of Eq. (12) one obtains

?Sz?

S

∆

√W2+ ∆2

∆?ˆ˜ σx? − W?ˆ˜ σz?

√W2+ ∆2

=

W

√W2+ ∆2

?|K−|2− |K+|2?

?K∗

.

+

+K−+ K+K∗

−

?

=(19)

B.Magneto-elastic equations

We want to describe our system of N spins in terms of

the spin field

ˆ n(r) =

N

?

i

ˆ˜ σiδ(r − ri), (20)

satisfying commutation relations

[ˆ nα(r), ˆ nβ(r′)] = 2iǫαβγˆ nγ(r)δ(r − r′).

In terms of this field the total Hamiltonian becomes

(21)

ˆ H = −1

2

?

d3r ˆ n(r) · b(eff)(r) +ˆ Hph.(22)

The classical pseudo-spin field n(r,t) can be defined as

n(r,t) = ?ˆ n(r)?,(23)

where ?...? contains the average over quantum spin states

and the statistical average over spins inside a small vol-

ume around the point r. If the size of that volume is small

compared to the wavelength of the phonon displacement

field, then, as has been discussed in the Introduction,

n2(r) should be approximately constant in time. Accord-

ing to equations (17), (19) and (20), the magnetization

is given by

Mz(r) = gµBS∆nx(r) − W nz(r)

√W2+ ∆2

.(24)

The dynamical equation for the classical pseudo-spin

field n(r,t) is

i?∂n(r,t)

∂t

=

?

[ˆ H, ˆ n]

?

,(25)

which, with the help of Eq.(21), can be written as

?∂n(r,t)

∂t

= n(r,t) × b(eff)(r,t).(26)

In this treatment we are making a common assumption

that averaging over spin and phonon states can be done

independently. This approximation is expected to be

good in the long-wave limit.

The dynamical equation for the displacement field is

ρ∂2uα

∂t2

=

?

β

∂σαβ

∂xβ

, (27)

where σαβ = ∂h/∂eαβ is the stress tensor, eαβ =

∂uα/∂xβis the strain tensor, h is the Hamiltonian den-

sity of the system inˆ H =?d3rh(r), and ρ is the mass

ric part originating from the magneto-elastic interaction

in the Hamiltonian,

density. Note that the stress tensor has an antisymmet-

σαβ = σ(s)

αβ+ σ(a)

1

2S∆ny(r)ǫzαβ.

αβ,

σ(a)

αβ=

(28)

This implies that at each point r there is a torque per

unit volume,

τα(r) = −δαzS∆ny(r),(29)

created by the interaction with the magnetic system.

This effect can be viewed as the local Einstein – de Haas

effect: Spin rotation produces a torque in the crystal lat-

tice due to the necessity to conserve angular momentum.

With the help of equations (16), (22), and (27), using

standard results of the theory of elasticity, one obtains

∂2uα

∂t2− c2

t∇2uα−(c2

l−c2

t)∇α(∇ · u) =S∆

2ρǫzαβ∇βny,

(30)

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where cland ctare velocities of longitudinal and trans-

verse sound. The source of deformation in the right hand

side of this equation is due to the above-mentioned torque

generated by the spin rotation.

Equations (26) and (30) describe coupled motion of

the pseudospin field n(r,t) and the displacement field

u(r,t). It is easy to see from these equations that in ac-

cordance with the argument presented in the Introduc-

tion n2

zis independent of time. It may, nev-

ertheless, depend on coordinates, reflecting the structure

of the initial state. In this paper we study cases in which

the crystal of molecular magnets was initially prepared

in the ground state n = n0ezwith n0being the concen-

tration of magnetic molecules. In this case the dynamics

of n(r) described by equations (26) and (30) reduces to

its rotation, with the length of n(r) being a constant n0.

Remarkably, this situation is similar to a ferromagnet,

despite the absence of the exchange interaction.

x+ n2

y+ n2

III.LINEAR MAGNETO-ELASTIC WAVES

A. Bulk waves

For magnetic molecules whose magnetic cores are more

rigid than their environments, only the transverse part

of the displacement field (with ∇ · u(r) = 0) interacts

with the magnetic degrees of freedom. This is a conse-

quence of the fact that the elastic deformation produced

by the rotation of local magnetization is a local twist of

the crystal lattice, required by the conservation of angu-

lar momentum. Let us consider then a transverse plane

wave propagating along the X-axis. From Eqs.(26) and

(30) one obtains

∂2uy

∂t2− c2

?∂nx

∂t

?∂ny

∂t

?∂nz

∂t

t

∂2uy

∂x2= −S∆

W2+ ∆2− nzS∆∂uy

?

= S∆nx∂uy

∂x

2ρ

∂ny

∂x

= ny

?

∂x

= −nx

W2+ ∆2

. (31)

We shall study linear waves around the ground state

|ψ+? corresponding to nz = n0,nx,y = 0,uy = 0. The

perturbation around this state results in nonzero nx,y

and uy. Linearized equations of motion are

∂2uy

∂t2− c2

?∂nx

∂t

?∂ny

∂t

t

∂2uy

∂x2= −S∆

W2+ ∆2− S∆n0∂uy

?

2ρ

∂ny

∂x

= ny

?

∂x

= −nx

W2+ ∆2. (32)

For uy,nx,y ∝ exp(iqx − iωt), the above equations be-

FIG. 1: Interacting sound and spin modes. Notice the gap

below spin resonance ω0.

come

(ω2− c2

iqn0S∆√W2+ ∆2

tq2)uy− iqS∆

2ρny= 0

?2

uy+

?

ω2−W2+ ∆2

?2

?

ny= 0.

(33)

The spectrum of coupled excitations is given by

(ω2− c2

tq2)

?

ω2−W2+ ∆2

?2

?

=n0S2∆2√W2+ ∆2

2ρ?2

q2.

(34)

In the vicinity of the resonance,

ctq0=

√W2+ ∆2

?

≡ ω0,(35)

one can write

ω = ω0(1 + δ)(36)

with δ to be determined by the dispersion relation. Sub-

stituting equations (35) and (36) into Eq. (34), one ob-

tains

δ = ±

?

n0S2∆2

8ρc2

t?ω0

, (37)

that describes the splitting of two coupled modes at the

resonance. The repulsion of elastic and spin modes is

illustrated in Fig. 1. The relative splitting of the modes

reaches maximum at W = 0 (?ω0= ∆):

2|δmax| =

?

n0S2∆

2ρc2

t

= S

?

∆

2Mc2

t

,(38)

where M = ρ/n0 is the mass of the volume containing

one molecule of spin S. Notice also another consequence

of Eq. (34): The presence of the energy gap below ω0=

√W2+ ∆2/? (see Fig. 1). The value of the gap follows

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from Eq. (34) at large q. It equals 2δ2ω0. This effect

is qualitatively similar to the one obtained in Ref.

from an ad hoc model of spin-phonon interaction.

contrast with that model our results for the splitting of

the modes and for the gap do not contain any unknown

interaction constants as they are uniquely determined by

the conservation of the total angular momentum (spin +

crystal lattice).

According to equations (33) and (34) the Fourier trans-

forms of nyand uyare related through

5

In

ny

n0

= iS

ω2

0− ω2

0

ω2

∆

?ω0

quy.(39)

Due to the condition of the elastic theory quy≪ 1, the

absolute value of the ratio ny/n0is generally small, un-

less ω is close to ω0. This means that away from the

resonance the sound cannot significantly change the pop-

ulation of excited spin states. At the magneto-elastic

resonance, substituting equations (36) and (37) into the

above equation, one obtains:

|ny|res

n0

=

?2Mω0

?

?1/2

|uy|.(40)

Although this relation is valid only at |ny| ≪ n0, it allows

one to estimate the amplitude of ultrasound that will

significantly affect populations of spin states. We shall

postpone the discussion of this effect until Section V.

Meantime let us compute the magnetization generated

by the linear elastic wave, uy = u0cos[q0(x − ctt)]), in

resonance with our two-state spin system. The last of

Eqs. (32) yields nx= i(ω/ω0)ny. Then, with the help of

Eq. (24) and Eq. (40) one obtains

Mz= gµBS∆

?ω0

?2Mc2

t

?ω0

?1/2

q0u0cos[q0(x − ctt)]. (41)

So far we have investigated coupled magneto-elastic

waves in the vicinity of the ground state, nz= n0. Eqs.

(31) also allow one to obtain the increment, Γ, of the

decay of the unstable macroscopic state of the crystal,

nz= −n0, in which all molecules are initially in the ex-

cited state |ψ−?. In fact, the result can be immediately

obtained from equations (32) – (34) by replacing n0with

−n0. It is then easy to see from Eq. (34) that in the vicin-

ity of the resonance the frequency acquires an imaginary

part that attains maximum at the resonance where

ω = ω0(1 ± i|δ|).(42)

The mode growing at the rate Γ = ω0|δ| represents the

decay of |ψ−? spin states into |ψ+? spin states, separated

by energy ?ω0. This decay is accompanied by the ex-

ponential growth of the amplitude of ultrasound of fre-

quency ω0.

B. Surface waves

Magneto-elastic coupling in crystals of molecular mag-

nets can be studied with the help of surface acoustic

FIG. 2: Geometry of the problem with surface acoustic waves.

waves (see Discussion). To describe the surface waves

we chose a geometry in which the surface of interest

is the XZ-plane and the solid extends to y > 0 with

waves running along the direction that makes an angle

θ with the X-axis, see Fig.2.

that the displacement field u(r,t) and the components

nx(r,t),ny(r,t) have the form

As usual22we assume

A = A0e−αyei(qxx+qzz)e−iωt.(43)

It is convenient to express the components of the dis-

placement field in the coordinate system defined by

(el,et,ep), see Fig.2,

ux = ulcosθ − utsinθ

uy = up

uz = ulsinθ + utcosθ.(44)

Equations of motion for ul, ut, and up follow from Eq.

(30):

?ω2+ c2

?ω2+ c2

t(α2− q2)?ut +

tα2− c2

S

2ρα∆sinθny= 0

l− c2

S

2ρα∆cosθny= 0

l− c2

iS

2ρ∆q cosθny= 0.

lq2?ul − iαq(c2

−

lα2− c2

t)up

?ω2+ c2

tq2?up − iαq(c2

−

t)ul

(45)

It is easy to see that for θ ?= kπ , k = 0,1,2... and ny?= 0,

the transverse component ut cannot be zero, contrary

to the case of Rayleigh waves. This is the signature of

magneto-elastic coupling.

As in the analysis of bulk waves, we shall study the

linear waves around the ground state corresponding to

the pseudospin field polarized in the Z-direction, nz =

n0,nx,y = 0. The excitations above this state are de-

scribed by Eqs. (26), which become

− i?ωnx = S∆?−α(ulcosθ − utsinθ) − iq?cosθup

+

?

−i?ωny = −

Substitution of these two equations into Eqs.(45) leads

to a homogeneous system of algebraic equations for ul,

?

W2+ ∆2ny

?

W2+ ∆2nx.(46)